Various questions around finitely positively expansive dynamical systems

It is well-known that when a positively expansive dynamical system is invertible then its underlying space is finite. C.Morales has introduced a decade ago a natural way to generalize positive expansiveness, by introducing other properties that he called positive $n$-expansiveness, for all $n \ge 1$, positive $1$-expansiveness being identical to positive expansiveness. Contrary to positive expansiveness, positive $n$-expansiveness for $n>1$ does not enforce that the space is finite when the system is invertible. In the present paper we call finitely positively expansive dynamical systems as the ones which are positively $n$-expansive for some integer $n$, and prove several results on this class of systems. In particular, the well-known result quoted above is true if we add the constraint of shadowing property, while it is not if this property is replaced with minimality. Furthermore, finitely positively expansive systems cannot occur on certain topological spaces such as the interval, when the system is assumed to be invertible finite positive expansiveness implies zero topological entropy. Overall we show that the class of finitely positively expansive dynamical systems is quite rich and leave several questions open for further research.